Special Children – Part 1

 Dr. Coplan lays the foundation for the concept of “special children.”

All children are special, but (borrowing from George Orwell*) some are more special than others. I don’t mean “special children” in the way that it’s often used: as a code term for “disabled.” To get at what I mean by “special children,” and why some children are “more special” than others, we need to do a bit of simple Math. Don’t panic!!! This is pretty straightforward. Honest!

Take a look at the first line in the box above. How many whole numbers are there from one through ten? Count them: One, two, three, etc., all the way to 10. (Don’t worry about the dots and that symbol on the right for the moment). Your answer should be: “There are ten whole numbers from one through ten.”

Now take a look at the lower line. These are the odd numbers that lie between one and ten. How many are there? Your answer should be “Five. There are five odd numbers that lie between one and ten: 1, 3, 5, 7, and 9 = five numbers.”

In other words, there are half as many odd numbers (five) as whole numbers (ten) between 1 and 10. Likewise, there are one hundred whole numbers between 1 and 100, but there are only fifty odd numbers. There are one thousand whole numbers between 1 and 1000, but only 500 odd numbers. There are one million whole numbers between 1 and 1,000,000, but only half a million odd numbers. And so on. So, we can make the following general statement: “Within any specified range of numbers, there will be twice as many whole numbers as odd numbers.” That’s just a formal way of saying something that is obvious when we stop to think about. Are you still with me? There is nothing tricky here. No hidden “Gotcha,” or difficult calculations. Just counting. So far, so good.

Now what about the row of dots, and that funny little symbol that looks sort of like an 8 lying on its side? That symbol () means “Infinity.” Like the Energizer Bunny, Infinity just keeps going and going and going….forever. So the first line in the box is telling us “Start with the number ‘one,’ and continue counting whole numbers, forever.” How many numbers is that? There is no such thing as “the last whole number in the universe.” They keep on going forever: 1, 2, 3, 4, 5, 6…. one billion…one trillion….etc. So the answer is: There is an infinite number of whole numbers.

Now look at the second row inside the box. The second row is telling us “Start with the number ‘one,’ and continue counting just the odd numbers, forever.” How many numbers is that? There is no such thing as “the last odd number in the universe.” Like whole numbers, the odd numbers keep going and going: 1, 3, 5, 7, 9, 11, 13….one billion and 1…..one trillion and 1, etc., forever. So, how many odd numbers are there in the universe? Answer: There is an infinite number of odd numbers.

Here comes the tricky part: At the beginning of this exercise, we showed that within any specified range of numbers, there are twice as many whole numbers as odd numbers (or, stated the other way, there are only half as many odd numbers as whole numbers – because the other half are the even numbers.) So, intuitively, it seems that there ought to be twice as many whole numbers as odd numbers in the universe, but both sets (the set of all whole numbers, and the set of all odd numbers) are infinitely large. If two sets are both infinitely large, how can one be “bigger” or “smaller” than the other? Yikes!

What I’ve done here, by asking you to do nothing more than count, is to expose a paradox in number theory. We needn’t bother ourselves with the math that’s required to address the paradox (click here and here ). But the paradox itself is our jumping off point for what I mean by special children: Intuitively, the set of all whole numbers seems to be “twice as large” as the set of all odd numbers, even though both are infinitely large. In a similar way, even though every child is infinitely special, some children are “more special” than others.

Next time we’ll talk about what makes some children “more special” than others — and what that may have to do with your family.

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*(The original line, from George Orwell’s 1945 novel, Animal Farm was “All animals are equal, but some are more equal than others.” The book was prescient, and I recommend it highly.)

James Coplan, MD is an Internationally recognized clinician, author, and public speaker in the fields of early child development, early language development and autistic spectrum disorders. Stay connected, join Dr. Coplan on Facebook and Twitter.